After a finite amount of time, it will either report "Halt" or "Infinite Loop", but my program won't take forever, just a finite amount of time.

then you have added nothing to the problem. of course if you run the program and it halts, even if runs until T+heatdeathofuniverse^googleplex (which is a finite time) without an infinite loop, you have not solved the conundrum implied by Turing’s work. Turing proved no algorithm exists that always correctly decides whether, for a given arbitrary program and input, the program halts when run with that input. The essence of Turing's proof is that any such algorithm can be made to produce contradictory output and therefore cannot be correct. He proved it - QED